Dissipativity of θ-methods for a class of nonlinear neutral delay integro-differential equations

References

• Alnasr, M. H. , 2004. The numerical stability of multistep methods for Volterra integral equations with many delays , Int. J. Comput. Math. 81 (2004), pp. 1257–1263.
   Web of Science ®Google Scholar
• Bellen, A. , and Zennaro, M. , 2003. Numerical Methods for Delay Differential Equations . Oxford: Oxford University Press; 2003.
   Google Scholar
• Bellen, A. , Guglielmi, N. , and Zennaro, M. , 2000. Numerical stability of nonlinear delay differential equations of neutral type , J. Comput. Appl. Math. 125 (2000), pp. 251–263.
   Web of Science ®Google Scholar
• Bocharov, G. A. , and Rihan, F. A. , 2000. Numerical modelling in biosciences using delay differential equations , J. Comput. Appl. Math. 125 (2000), pp. 183–199.
   Web of Science ®Google Scholar
• Brayton, R. K. , 1968. Small signal stability criterion for networks containing lossless transmission lines , IBM J. Res. Dev. 12 (1968), pp. 431–440.
   Web of Science ®Google Scholar
• Brunner, H. , 1994. The numerical solutions of neutral Volterra integro-differential equations with delay arguments , Ann. Numer. Math. 1 (1994), pp. 309–322.
   Google Scholar
• Brunner, H. , 2004. "Collocation Methods for Volterra Integral and Related Functional Differential Equations". In: Cambridge Monographs on Applied and Computational Mathematics . Vol. 15. Cambridge: Cambridge University Press; 2004.
   Google Scholar
• Dahlquist, G. , 1976. "Error analysis for a class of methods for stiff non-linear initial value problems". In: Watson, G. A. , ed. Numerical Analysis, Dundee, 1975 . Berlin: Springer-Verlag; 1976. pp. 60–72, Lecture Notes in Mathematics Vol. 506.
   Google Scholar
• Enright, W. H. , and Hu, M. , 1997. Continuous Runge–Kutta methods for neutral Volterra integro-differential equations with delay , Appl. Numer. Math. 24 (1997), pp. 175–190.
   Web of Science ®Google Scholar
• Gan, S. Q. , 2006. Dissipativity of linear θ-methods for integro-differential equations , J. Comput. Math. Appl. 52 (2006), pp. 449–458.
   Web of Science ®Google Scholar
• Gan, S. Q. , 2007. Dissipativity of θ-methods for nonlinear Volterra delay integro-differential equations , J. Comput. Appl. Math. 206 (2007), pp. 898–907.
   Web of Science ®Google Scholar
• Gan, S. Q. , 2009. Dissipativity of θ-methods for nonlinear delay differential equations of neutral type , Appl. Numer. Math. 59 (2009), pp. 1354–1365.
   Web of Science ®Google Scholar
• Hale, J. K. , 1993. Theory of Functional Differential Equations . New York: Spring-Verlag; 1993.
   Google Scholar
• Hu, P. , and Huang, C. M. , 2011. Analytical and numerical stability of nonlinear neutral delay integro-differential equations , J. Franklin Inst. 348 (2011), pp. 1082–1100.
   Web of Science ®Google Scholar
• Huang, C. M. , 2000. Dissipativity of Runge–Kutta methods for dynamical systems with delays , IMA J. Numer. Anal. 20 (2000), pp. 153–166.
   Web of Science ®Google Scholar
• Huang, C. M. , 2000. Dissipativity of one-leg methods for dynamical systems with delays , Appl. Numer. Math. 35 (2000), pp. 11–22.
   Web of Science ®Google Scholar
• Huang, C. M. , and Chang, Q. S. , 2004. Dissipativity of multistep Runge–Kutta methods for dynamical systems with delays , Math. Comput. Model. 40 (2004), pp. 1285–1296.
   Web of Science ®Google Scholar
• Huang, C. M. , and Chen, G. N. , 2001. Dissipativity of θ-methods for delay dynamical systems , Chinese J. Numer. Math. Appl. 23 (1) (2001), pp. 108–114.
   Google Scholar
• Humphries, A. R. , and Stuart, A. M. , 1994. Runge–Kutta methods for dissipative and gradient dynamical systems , SIAM J. Numer. Anal. 31 (1994), pp. 1452–1485.
   Web of Science ®Google Scholar
• Jerri, A. , 1999. "Introduction to Integral Equations with Applications". New York: Wiley; 1999.
   Google Scholar
• Kolmanovskii, V. , and Myshkis, A. , 1999. Introduction to the Theory and Applications of Functional Differential Equations . Dordrecht: Kluwer Academic Publishers; 1999.
   Google Scholar
• Koto, T. , 2002. Stability of Runge–Kutta methods for delay integro-differential equations , J. Comput. Appl. Math. 145 (2002), pp. 483–492.
   Web of Science ®Google Scholar
• Liu, X. Y. , and Wen, L. P. , 2010. Dissipativity of one-leg methods for neutral delay integro-differential equations , J. Comput. Appl. Math. 235 (2010), pp. 165–173.
   Web of Science ®Google Scholar
• Mahmouda, M. S. , Shi, Y. , and Sunni, F. M. , 2009. Dissipativity analysis and synthesis of a class of nonlinear systems with time-varying delays , J. Franklin Inst. 346 (2009), pp. 570–592.
   Web of Science ®Google Scholar
• Pinello, W. , Cangellaris, A. C. , and Ruehli, A. , 1997. Hybrid electromagnetic modeling of noise interactions in packaged electronics based on the partial element equivalent circuit formulation , IEEE Trans. Microw. Theory Tech. 45 (1997), pp. 1889–1896.
   Web of Science ®Google Scholar
• Song, Q. K. , and Zhao, Z. J. , 2005. Global dissipativity of neural networks with both variable and unbounded delays , Chaos Soliton Fractals 25 (2005), pp. 393–401.
   Web of Science ®Google Scholar
• Stuart, A. M. , and Humphries, A. R. , 1998. Dynamical Systems and Numerical Analysis . Cambridge: Cambridge University Press; 1998.
   Google Scholar
• Temam, R. , 1988. "Infinite-Dimensional Dynamical Systems in Mechanics and Physics". In: Springer Applied Mathematical Sciences Series . Vol. 68. Berlin: Springer; 1988.
   Google Scholar
• Tian, H. J. , 2004. Numerical and analytic dissipativity of the θ-method for delay differential equation with a bounded variable lag , Int. J. Bifur. Chaos Appl. Sci. Eng. 14 (2004), pp. 1839–1845.
   Web of Science ®Google Scholar
• Tian, H. J. , and Guo, N. , 2008. Asymptotic stability, contractivity and dissipativity of one-leg θ-method for non-autonomous delay functional differential equations , Appl. Math. Comput. 202 (2008), pp. 780–786.
   Web of Science ®Google Scholar
• Tian, H. J. , Fan, L. Q. , and Xiang, J. X. , 2007. Numerical dissipativity of multistep methods for dynamical systems with delays , Appl. Math. Comput. 188 (2007), pp. 934–941.
   Web of Science ®Google Scholar
• Tian, H. J. , Guo, N. , and Shen, A. L. , 2009. Dissipativity of delay functional differential equations with a bounded lag , J. Math. Anal. Appl. 355 (2009), pp. 778–782.
   Web of Science ®Google Scholar
• Wang, W. S. , 2010. A generalized Halanay inequality for stability of nonlinear neutral functional differential equations , J. Inequal. Appl. 2010 (2010), pp. 1–16, article 475019.
   Google Scholar
• Wang, W. S. , and Li, S. F. , 2008. Dissipativity of Runge–Kutta methods for neutral delay differential equations with piecewise constant delay , Appl. Math. Lett. 21 (2008), pp. 983–991.
   Web of Science ®Google Scholar
• Wen, L. P. , and Li, S. F. , 2006. Dissipativity of Volterra functional differential equations , J. Math. Anal. Appl. 324 (2006), pp. 696–706.
   Web of Science ®Google Scholar
• Wen, L. P. , Wang, W. S. , and Yu, Y. X. , 2008. Dissipativity of θ-method for a class of nonlinear neutral delay differential equations , Appl. Math. Comput. 202 (2008), pp. 780–786.
   Web of Science ®Google Scholar
• Wen, L. P. , Yu, Y. X. , and Wang, W. W. , 2008. Generalized Halanay inequalities for dissipativity of Volterra functional differential equations , J. Math. Anal. Appl. 347 (2008), pp. 169–178.
   Web of Science ®Google Scholar
• Wen, L. P. , Wang, S. X. , and Yu, Y. X. , 2009. Dissipativity of Runge–Kutta methods for neutral delay integro-differential equations , Appl. Math. Comput. 215 (2009), pp. 583–590.
   Web of Science ®Google Scholar
• Wen, L. P. , Wang, W. W. , and Yu, Y. X. , 2010. Dissipativity and asymptotical stability of nonlinear neutral delay integro-differential equations , Nonlinear Anal. 72 (2010), pp. 1746–1754.
   Web of Science ®Google Scholar
• Wu, S. F. , and Gan, S. Q. , 2008. Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations , Comput. Math. Appl. 55 (2008), pp. 2426–2443.
   Web of Science ®Google Scholar
• Yu, Y. X. , Wen, L. P. , and Li, S. F. , 2007. Nonlinear stability of Runge–Kutta methods for neutral delay integro-differential equations , Appl. Math. Comput. 191 (2007), pp. 543–549.
   Web of Science ®Google Scholar
• Zhang, C. J. , and Li, S. F. , 2001. Dissipativity and exponentially asymptotic stability of the solution for nonlinear neutral functional differential equations , Appl. Math. Comput. 119 (2001), pp. 109–115.
   Web of Science ®Google Scholar
• Zhang, C. J. , and Vandewalle, S. , 2004. Stability analysis of Volterra delay-integro-differential equations and their backward differentiation time discretization , J. Comput. Appl. Math. 164–165 (2004), pp. 797–814.
   Web of Science ®Google Scholar
• Zhang, C. J. , and Vandewalle, S. , 2004. Stability analysis of Runge–Kutta methods for nonlinear Volterra delay-integro-differential equations , IMA J. Numer. Anal. 24 (2004), pp. 193–214.
   Web of Science ®Google Scholar
• Zhang, C. J. , and Vandewalle, S. , 2006. General linear methods for Volterra integro-differential equations with memory , SIAM J. Sci. Comput. 27 (2006), pp. 2010–2031.
   Web of Science ®Google Scholar
• Zhang, C. J. , and Vandewalle, S. , 2007. Stability criteria for exact and discrete solutions of neutral multidelay integro-differential equations , Adv. Comput. Math. 28 (2007), pp. 383–399.
   Web of Science ®Google Scholar
• Zhao, J. J. , Xu, Y. , and Liu, M. Z. , 2005. Stability analysis of numerical methods for linear neutral Volterra delay integro-differential equations , Appl. Math. Comput. 167 (2005), pp. 1062–1079.
   Web of Science ®Google Scholar
• Zhen, C. , and Huang, C. M. , 2007. Dissipativity for nonlinear neutral delay differential equations , J. Syst. Simul. 19 (2007), pp. 3184–3187, (in Chinese).
   Google Scholar